CMC A-net surfaces in three dimensional Heisenberg group

In this paper, we obtain the mean curvature of a Anet surface in three dimensional Heisenberg group H3. Moreover, we give some characterizations of this surface according to LeviC i ita connections of H3. Using the mean curvature, a new characterization for the cmc Anet surface. Finally, we draw cmc Anet surface by Mathematica.


Introduction
The study of mean curvature expands back to 18. century. Later, Lagrange looked for a necessary condition to minimizing a certain integral then, he found the minimal surface equation. Meusnier firstly define the term of mean curvature. Then, important mathematicians such as Gauss and Weierstrass devoted much of their studies to these surfaces. Finally, constant mean curvature surfaces remain somewhat mysterious to this day.
The significance of CMC hypersurfaces as useful tools for studying the physics of general relativistic spacetimes is widely recognized. Nevertheless little is yet known about the class of spacetimes which admit them. It has been conjectured that every maximally extended, globally hyperbolic, spatially compact solution of Einstein's equations (in vacuum or with "reasonable" source field coupling) can be foliated by CMC hypersurfaces. However this conjecture is known to be true only for a handful of examples such as the spatially homogeneous cosmological models, [7]. Also, The partial differential equation H = const. can be considered the Euler equation to the variational problem.
In [16], Soyuçok interested with the Bonnet problem of determining the surfaces in Euclidean three dimensional space which can admit at least one nontrivial isometry that preserves principal curvatures. This problem considered locally and examined the general case. Then, In [8], Kanbay considered the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures, then, she determined the Bonnet ruled surfaces whose generators and orthogonal trajectories form a special net called an A-net.
In [4], Brander studied constant positive Gauss curvature K surfaces in the 3-sphere S 3 with 0 < K < 1 as well as constant negative curvature surfaces. They showed that the so-called normal Gauss map for a surface in S 3 with Gauss curvature K < 1 is Lorentz harmonic with respect to the metric induced by the second fundamental form if and only if K is constant. They gave a uniform loop group formulation for all such surfaces with K = 0, and use the generalized d'Alembert method to construct examples. Then, they obtain that those representation gives a natural correspondence between such surfaces with K < 0 and those with 0 < K < 1. In [6], Galvez deal with some classical results on complete surfaces with constant Gauss curvature in 3-dimensional space forms with using a modern approach. Then, he obtain characterizetion on the complete surfaces with positive constant Gauss curvature in S 3 , R 3 or H 3 and proved the Liebmann theorem and show that the only complete examples must be totally umbilical round spheres. He deduce that there is no complete surface in S 3 with constant Gauss curvature K (I) ∈ (0, 1). In [12], Lopez studied a parabolic surface in hyperbolic space H 3 is a surface invariant by a group of parabolic isometries, then, describe all parabolic surfaces with constant Gaussian curvature. In [13], Lopez gave a space-like or time-like surface in Lorentz-Minkowski three-space L 3 generated by a oneparameter family of circular arcs and later, he obtain if the Gauss curvature K is a nonzero constant, then M is a surface of revolution. We also describe the parametrizations for M when K ≡ 0.
In this paper, we studied A-net surfaces in three dimensional Heisenberg group. Then, we made some characterizations of curvatures of these surfaces.

Heisenberg Group H 3
The Heisenberg group historically originates in and still has its strongest ties to quantum physics: there it is a group of unitary operators acting on the space of states induced from those observables on a linear phase space, which are given by linear or by constant functions. So any Heisenberg group is a subgroup of a group of observables in certain simple examples of quantum mechanical systems.
The Heisenberg group H 3 is defined as R 3 with the group operation The left-invariant Riemannain metric on H 3 is given by The left invariant orthonormal frame on H 3 , which is belong to Riemannian metric g For the covariant derivatives of the Levi-Civita connection of the left-invariant metric g, where the (i, j)-element in the table above equals ∇ e i e j for our basis. Also, we have the Heisenberg bracket relations Let be a surface in (H 3 , g) . If we take derivatives of the surface, which is given with the parametrization (1), we have ϕ y (x, y) = ξ ′ (y) e 2 + ρ y (x, y) e 3 .

Theorem 1.
Let ϕ (x, y) be a surface which is parameterized as (1). If ρ y = 0, then the mean curvature of the surface ϕ (x, y) Proof. From equations (2), we have Then, components of the first fundamental form of the surface ϕ (x, y) are

So, the induced metric isg
The unit normal vector field of the surface is Then, we have So, from (7) and (8) Then, the mean curvature of the surface ϕ (x, y) Theorem 2. Let ϕ (x, y) be a surface which is parameterized as (1). If ϕ (x, y) is an A-net cmc surface with ρ y = 0, then Proof. From (13), (14), we have and where C = A B . Then, from (18)-(22), the mean curvature is So, if the mean curvature of the surface is a constant, from (21), we have Derivative of the equation (22), we have So, from equation (23), In this situation, we have two possibilies.
(i) If C = 0, then we have h 12 = 0, which is conrtadicts with the surface is a A-net surface.
Then, if we notice that α ′ (x) = P (x) , the equation (25) become So, we have With integrate of the equation (27) according to x, we have where c 1 and c 2 are integration constant. So, if ϕ (x, y) is a cmc A-net surface, we have So, the proof is complete. Proof. From equations (6), (12)- (15), the Gaussian curvature is ϕ (x, y) has constant Gaussian curvature.
Proof. The proof obtains like Theorem 2.
This surface is a A-net cmc surface in (H 3 , g).